Proposition 8

Let E be a separated locally convex space and $u$ an endomorphism of E. The following conditions are equivalent:

(i) $u$ is a Riesz endomorphism of E;

(ii) There exists a quasi-inverse $v$ of $u$ which commutes with $u$;

(iii) There exists a closed vector subspace $E_1$ of E, of finite codimension, stable under $u$, such that $u$ induces an automorphism of $E_1$.

The implication (i) $=\Rightarrow$ (ii) follows from prop. 6, d).

Suppose that $u$ possesses a quasi-inverse $v$ which is permutable with $u$. The kernel $E_1$ of $1_E-u\circ v$ is a closed vector subspace of E, of finite codimension, stable under $u$ and $v$. Since one has $u(v(x)) =v(u(x)) =x$ for every $x\in E_1$, the mappings $u$ and $v$ induce automorphisms of $E_1$ inverse to one another. Thus (ii) implies (iii).

Finally, the implication (iii) $=\Rightarrow$ (i) follows from assertion a) of prop. 7 and from the fact that an automorphism is a Riesz endomorphism.